Geometrical Applications of Complex Numbers

Interactive Feed

JEE Main 2014

LEVELJEE Main

If $z$ is a complex number such that $∣ z ∣ \geq 2$ , then the minimum value of $∣ z + 1/2∣$

(A)

is strictly greater than 5/2

(B)

is strictly greater than 3/2 but less than 5/2

(C)

is equal to 5/2

(D)

lie in the interval (1, 2)

JEE Main 2012

LEVELJEE Main

If $z \neq = 1$ and $\frac{z ^{2}}{z - 1}$ is real, then the point represented by the complex number $z$ lies

(A)

either on the real axis or on a circle passing through the origin.

(B)

on a circle with centre at the origin

(C)

either on the real axis or on a circle not passing through the origin.

(D)

on the imaginary axis.

JEE Main 2011

LEVELJEE Main

Let $α, β$ be real and $z$ be a complex number. If $z^{2} + α z + β = 0$ has two distinct roots on the line $R ez = 1$ , then it is necessary that

(A)

β \in (- 1, 0)

(B)

β ∣ = 1

(C)

β \in (1, \infty)

(D)

β \in (0, 1)

JEE Main 2010

LEVELJEE Main

The number of complex numbers $z$ such that $∣ z - 1∣ = ∣ z + 1∣ = ∣ z - i ∣$ equals

(A)

(B)

(C)

\infty

(D)

JEE Main 2007

LEVELJEE Main

If $∣ z + 4∣ \leq 3$ , then the maximum value of $∣ z + 1∣$ is

If $z_{1}$ and $z_{2}$ are two non-zero complex numbers such that $∣ z_{1} + z_{2} ∣ = ∣ z_{1} ∣ + ∣ z_{2} ∣$ , then $a r g z_{1} - a r g z_{2}$ is equal to

(A)

π /2

(B)

- π

(C)

(D)

- π /2

JEE Main 2005

LEVELJEE Main

If $ω = \frac{z}{z - \frac{1}{3} i}$ and $∣ ω ∣ = 1$ , then $z$ lies on

If $∣ z^{2} - 1∣ = ∣ z ∣^{2} + 1$ , then $z$ lies on

Let $Z_{1}$ and $Z_{2}$ be two roots of the equation $Z^{2} + a Z + b = 0$ , $Z$ being complex. Further, assume that the origin, $Z_{1}$ and $Z_{2}$ form an equilateral triangle. Then

(A)

a^{2} = 4 b

(B)

a^{2} = b

(C)

a^{2} = 2 b

(D)

a^{2} = 3 b

JEE Main 2002

LEVELJEE Main

If $∣ z - 4∣ < ∣ z - 2∣$ , its solution is given by

The locus of the centre of a circle which touches the circle $∣ z - z_{1} ∣ = a$ and $∣ z - z_{2} ∣ = b$ externally ( $z, z_{1}$ & $z_{2}$ are complex numbers) will be

Let $a, b \in R$ and $a^{2} + b^{2} \neq = 0$ . Suppose $S = {z \in C : z = \frac{1}{a + ib t}, t \in R, t \neq = 0}$ , where $i = - 1$ . If $z = x + i y$ and $z \in S$ , then $(x, y)$ lies on

* Multiple Correct Options

(A)

the circle with radius

1/2 a

and centre

(1/2 a, 0)

for

a > 0, b \neq = 0

(B)

the circle with radius

- 1/2 a

and centre

(- 1/2 a, 0)

for

a < 0, b \neq = 0

(C)

the x-axis for

a \neq = 0, b = 0

(D)

the y-axis for

a = 0, b \neq = 0

JEE Advanced 2013

LEVELJEE Advanced

Let complex numbers $α$ and $1/ \overset{α}{ˉ}$ lie on circles $(x - x_{0})^{2} + (y - y_{0})^{2} = r^{2}$ and $(x - x_{0})^{2} + (y - y_{0})^{2} = 4 r^{2}$ respectively. If $z_{0} = x_{0} + i y_{0}$ satisfies the equation $2∣ z_{0} ∣^{2} = r^{2} + 2$ , then $∣ α ∣ =$

(A)

1/ 2

(B)

1/2

(C)

1/ 7

Let $z_{1}$ and $z_{2}$ be two distinct complex numbers and let $z = (1 - t) z_{1} + t z_{2}$ for some real number $t$ with $0 < t < 1$ .

* Multiple Correct Options

(A)

∣ z - z_{1} ∣ + ∣ z - z_{2} ∣ = ∣ z_{1} - z_{2} ∣

(B)

A r g (z - z_{1}) = A r g (z - z_{2})

(C)

z - z_{1} z_{2} - z_{1} \overset{z}{ˉ} - \overset{z}{ˉ}_{1} \overset{z}{ˉ}_{2} - \overset{z}{ˉ}_{1} = 0

(D)

A r g (z - z_{1}) = A r g (z_{2} - z_{1})

JEE Advanced 2009

LEVELJEE Main

Let $z = x + i y$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation $z \overset{z}{ˉ}^{3} + \overset{z}{ˉ} z^{3} = 350$ is

A particle $P$ starts from the point $z_{0} = 1 + 2 i$ , where $i = - 1$ . It moves horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $z_{1}$ . From $z_{1}$ the particle moves $2$ units in the direction of the vector $\hat{i} + \hat{j}$ and then it moves through an angle $\frac{π}{2}$ in anticlockwise direction on a circle with centre at origin, to reach a point $z_{2}$ . The point $z_{2}$ is given by

(A)

6 + 7 i

(B)

- 7 + 6 i

(C)

7 + 6 i

(D)

- 6 + 7 i

JEE Advanced 2007

LEVELJEE Main

A man walks a distance of 3 units from the origin towards the north-east ( $N 4 5^{\circ} E$ ) direction. From there, he walks a distance of 4 units towards the north-west ( $N 4 5^{\circ} W$ ) direction to reach a point $P$ . Then the position of $P$ in the Argand plane is

(A)

3 e^{iπ /4} + 4 i

(B)

(3 - 4 i) e^{iπ /4}

(C)

(4 + 3 i) e^{iπ /4}

(D)

(3 + 4 i) e^{iπ /4}

JEE Advanced 2007

LEVELJEE Main

If $∣ z ∣ = 1$ and $z \neq = \pm 1$ , then all the values of $\frac{z}{1 - z ^{2}}$ lie on

(A)

a line not passing through the origin

(B)

∣ z ∣ = 2

The locus of $z$ which lies in shaded region (excluding the boundaries) is best represented by

(A)

z : ∣ z + 1∣ > 2

and

∣ a r g (z + 1) ∣ < π /4

(B)

z : ∣ z - 1∣ > 2

and

∣ a r g (z - 1) ∣ < π /4

(C)

z : ∣ z + 1∣ < 2

and

∣ a r g (z + 1) ∣ < π /2

(D)

z : ∣ z - 1∣ < 2

and

∣ a r g (z + 1) ∣ < π /2

JEE Advanced 2002S

LEVELJEE Main

For all complex numbers $z_{1}, z_{2}$ satisfying $∣ z_{1} ∣ = 12$ and $∣ z_{2} - 3 - 4 i ∣ = 5$ , the minimum value of $∣ z_{1} - z_{2} ∣$ is

The complex numbers $z_{1}, z_{2}$ and $z_{3}$ satisfying $\frac{z _{1} - z _{3}}{z _{2} - z _{3}} = \frac{1 - i 3}{2}$ are the vertices of a triangle which is

(A)

of area zero

(B)

right-angled isosceles

(C)

equilateral

(D)

obtuse-angled isosceles

JEE Advanced 1987

LEVELJEE Main

If $z_{1}$ and $z_{2}$ are two nonzero complex numbers such that $∣ z_{1} + z_{2} ∣ = ∣ z_{1} ∣ + ∣ z_{2} ∣$ , then $A r g z_{1} - A r g z_{2}$ is equal to

(A)

- π

(B)

- π /2

(C)

(D)

π /2

(E)

π

JEE Advanced 1986

LEVELJEE Main

Let $z_{1}$ and $z_{2}$ be complex numbers such that $z_{1} \neq = z_{2}$ and $∣ z_{1} ∣ = ∣ z_{2} ∣$ . If $z_{1}$ has positive real part and $z_{2}$ has negative imaginary part, then $(z_{1} + z_{2}) / (z_{1} - z_{2})$ may be

* Multiple Correct Options

If $a, b, c$ and $u, v, w$ are complex numbers representing the vertices of two triangles such that $c = (1 - r) a + r b$ and $w = (1 - r) u + r v$ , where $r$ is a complex number, then the two triangles

If $z_{1} = a + ib$ and $z_{2} = c + i d$ are complex numbers such that $∣ z_{1} ∣ = ∣ z_{2} ∣ = 1$ and $R e (z_{1} \overset{z}{ˉ}_{2}) = 0$ , then the pair of complex numbers $w_{1} = a + i c$ and $w_{2} = b + i d$ satisfies

* Multiple Correct Options

(A)

∣ w_{1} ∣ = 1

(B)

∣ w_{2} ∣ = 1

(C)

R e (w_{1} \overset{w}{ˉ}_{2}) = 0

(D)

none of these

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z is a complex number such that ∣z∣≥2, then the minimum value of ∣z+1/2∣

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z=1 and z−1z2​ is real, then the point represented by the complex number z lies

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α,β be real and z be a complex number. If z2+αz+β=0 has two distinct roots on the line Rez=1, then it is necessary that

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The number of complex numbers z such that ∣z−1∣=∣z+1∣=∣z−i∣ equals

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∣z+4∣≤3, then the maximum value of ∣z+1∣ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z1​ and z2​ are two non-zero complex numbers such that ∣z1​+z2​∣=∣z1​∣+∣z2​∣, then argz1​−argz2​ is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ω=z−31​iz​ and ∣ω∣=1, then z lies on

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∣z2−1∣=∣z∣2+1, then z lies on

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let Z1​ and Z2​ be two roots of the equation Z2+aZ+b=0, Z being complex. Further, assume that the origin, Z1​ and Z2​ form an equilateral triangle. Then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∣z−4∣<∣z−2∣, its solution is given by

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The locus of the centre of a circle which touches the circle ∣z−z1​∣=a and ∣z−z2​∣=b externally (z,z1​ & z2​ are complex numbers) will be

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b∈R and a2+b2=0. Suppose S={z∈C:z=a+ibt1​,t∈R,t=0}, where i=−1​. If z=x+iy and z∈S, then (x,y) lies on

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z1​ and z2​ be two distinct complex numbers and let z=(1−t)z1​+tz2​ for some real number t with 0<t<1.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let z=x+iy be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation zzˉ3+zˉz3=350 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∣z∣=1 and z=±1, then all the values of 1−z2z​ lie on

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The locus of z which lies in shaded region (excluding the boundaries) is best represented by

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For all complex numbers z1​,z2​ satisfying ∣z1​∣=12 and ∣z2​−3−4i∣=5, the minimum value of ∣z1​−z2​∣ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The complex numbers z1​,z2​ and z3​ satisfying z2​−z3​z1​−z3​​=21−i3​​ are the vertices of a triangle which is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If z1​ and z2​ are two nonzero complex numbers such that ∣z1​+z2​∣=∣z1​∣+∣z2​∣, then Argz1​−Argz2​ is equal to

If $z$ is a complex number such that $∣ z ∣ \geq 2$ , then the minimum value of $∣ z + 1/2∣$

If $z \neq = 1$ and $\frac{z ^{2}}{z - 1}$ is real, then the point represented by the complex number $z$ lies

Let $α, β$ be real and $z$ be a complex number. If $z^{2} + α z + β = 0$ has two distinct roots on the line $R ez = 1$ , then it is necessary that

The number of complex numbers $z$ such that $∣ z - 1∣ = ∣ z + 1∣ = ∣ z - i ∣$ equals

If $∣ z + 4∣ \leq 3$ , then the maximum value of $∣ z + 1∣$ is

If $z_{1}$ and $z_{2}$ are two non-zero complex numbers such that $∣ z_{1} + z_{2} ∣ = ∣ z_{1} ∣ + ∣ z_{2} ∣$ , then $a r g z_{1} - a r g z_{2}$ is equal to

If $ω = \frac{z}{z - \frac{1}{3} i}$ and $∣ ω ∣ = 1$ , then $z$ lies on

If $∣ z^{2} - 1∣ = ∣ z ∣^{2} + 1$ , then $z$ lies on

Let $Z_{1}$ and $Z_{2}$ be two roots of the equation $Z^{2} + a Z + b = 0$ , $Z$ being complex. Further, assume that the origin, $Z_{1}$ and $Z_{2}$ form an equilateral triangle. Then

If $∣ z - 4∣ < ∣ z - 2∣$ , its solution is given by

The locus of the centre of a circle which touches the circle $∣ z - z_{1} ∣ = a$ and $∣ z - z_{2} ∣ = b$ externally ( $z, z_{1}$ & $z_{2}$ are complex numbers) will be

Let $a, b \in R$ and $a^{2} + b^{2} \neq = 0$ . Suppose $S = {z \in C : z = \frac{1}{a + ib t}, t \in R, t \neq = 0}$ , where $i = - 1$ . If $z = x + i y$ and $z \in S$ , then $(x, y)$ lies on

Let $z_{1}$ and $z_{2}$ be two distinct complex numbers and let $z = (1 - t) z_{1} + t z_{2}$ for some real number $t$ with $0 < t < 1$ .

Let $z = x + i y$ be a complex number where $x$ and $y$ are integers. Then the area of the rectangle whose vertices are the roots of the equation $z \overset{z}{ˉ}^{3} + \overset{z}{ˉ} z^{3} = 350$ is

If $∣ z ∣ = 1$ and $z \neq = \pm 1$ , then all the values of $\frac{z}{1 - z ^{2}}$ lie on

The locus of $z$ which lies in shaded region (excluding the boundaries) is best represented by

For all complex numbers $z_{1}, z_{2}$ satisfying $∣ z_{1} ∣ = 12$ and $∣ z_{2} - 3 - 4 i ∣ = 5$ , the minimum value of $∣ z_{1} - z_{2} ∣$ is

The complex numbers $z_{1}, z_{2}$ and $z_{3}$ satisfying $\frac{z _{1} - z _{3}}{z _{2} - z _{3}} = \frac{1 - i 3}{2}$ are the vertices of a triangle which is

If $z_{1}$ and $z_{2}$ are two nonzero complex numbers such that $∣ z_{1} + z_{2} ∣ = ∣ z_{1} ∣ + ∣ z_{2} ∣$ , then $A r g z_{1} - A r g z_{2}$ is equal to